SISSA/101/2007/EP PreprinttypesetinJHEPstyle-HYPERVERSION hep-th/yymm.nnnn Spectral properties of ghost Neumann matrices 8 0 0 2 n L.Bonora a International School for Advanced Studies (SISSA/ISAS) J 4 Via Beirut 2–4, 34014 Trieste, Italy, and INFN, Sezione di Trieste 1 E-mail: [email protected], ] R.J.Scherer Santos h t - Centro Brasileiro de Pesquisas Fisicas (CBPF-MCT)-LAFEX p R. Dr. Xavier Sigaud, 150 - Urca - Rio de Janeiro - Brasil - 22290-180 e h E-mail: [email protected], [ 1 D.D.Tolla v Center for Quantum SpaceTime (CQUEST), Sogang University 9 9 Shinsu-dong 1, Mapo-gu, Seoul, Korea 0 E-mail: [email protected] 2 . 1 0 Abstract: We continue the analysis of the ghost wedge states in the oscillator formalism 8 0 bystudyingthespectralpropertiesoftheghostmatrices ofNeumanncoefficients. Weshow : v that the traditional spectral representation is not valid for these matrices and propose a Xi new heuristic formula that allows one to reconstruct them from the knowledge of their r eigenvalues and eigenvectors. It turns out that additional data, which we call boundary a data, areneededin ordertoactually implementthereconstruction. Inparticular ourresult lendssupporttotheconjecturethatthereexistsaghostthreestringsvertexwithproperties parallel to those of the matter three strings vertex. Keywords: String Field Theory, Ghost Wedge States, Star Product. Contents 1. Introduction 1 2. The three strings vertex 3 2.1 Ghost Neumann coefficients and their properties 5 2.2 Formulas for wedge states 6 2.3 Commutation relations with K 9 1 3. The diagonal recursive relations for wedge states 10 3.1 Proof of the diagonal recursive relations for wedge states 11 4. Matrix reconstruction from the spectrum 13 4.1 The problem 15 4.2 The solution 17 5. Conclusion 19 A. The ghost Neumann coefficients 20 B. Why we can use long square matrices 25 C. Proof of eq.(3.5) 26 1. Introduction This paper is complementary to the analysis, started in [1], of the conjectured equivalence e−n−22“L(0g)+L(0g)†” 0 = ec†Snb† 0 n (1.1) n | i N | i ≡ | i which is a crucial one in the recent developments in open string field theory [2, 3, 4, 5, 6, 7, 8, 10, 11, 12, 9, 13, 14]. Here n are the ghost wedge states in the oscillator formalism | i [15, 16, 17, 18, 19] , which of course must coincide with the corresponding surface wedge states. In [1] we dealt with the LHS of this equation. We showed that, if we understand it ordered according to the natural normal ordering, it can be cast into the midterm form in (1.1), and we diagonalized the matrix S in such a squeezed state. Then we proved that, n if we are allowed to star–multiply the squeezed states representing the ghost wedge states n the same way we do for the matter wedge states and diagonalize the corresponding | i matrices, the eigenvalue we obtain in the two cases are the same. In this paper we focus on the spectral properties of such operators like S and, among other things, we exhibit n evidence that the above if is justified. – 1 – To be more precise, in order to fully prove (1.1) we have two possibilities. The most direct alternative is to define the three strings vertex for the ghost part, and thus the star product, pertinent to the natural normal ordering in the oscillator formalism; then to construct the wedge states appropriate for this vertex; finally to diagonalize the latter and showsthattheycoincidewiththemidtermof(1.1)(withsomeadditionalspecificationsthat will be clarified in due course). Unfortunately the construction of the ghost vertex is not so straightforward as one would hope. Relying on the common lore on this subject, we face a large number of possibilities, which are mostly linked to the ghost zero mode insertions and our attempts in this direction so far have been unfruitful. Before continuing in such a challenging program it is wise to gather some evidence that the vertex one is looking for does exist and some indirect information about it. This is the original motivation of the present paper, which relies on the second alternative. Having diagonalized the matrices S in the midterm of (1.1) in the basis of weight 2 n differentials, see[1],onemaywonderwhetheronecanreconstructtheoriginalmatrices. For the matter part this is a standard procedure, simply one uses the spectral representation of the infinite matrices involved. But for the ghost sector we are interested in here things are morecomplicated (it shouldberecalled thatthe infinitematrices S are notsquarebut n lame, i.e. infinite rectangular). Ultimately, the answer is: yes, we can reconstruct the S n matrices; in other words, we can derive the RHS of (1.1) from the LHS, but the procedure is more involved than in the matter case. In fact the traditional spectral representation is not valid for lame matrices and we have to figure out a new heuristic formula that allows us to reconstruct them from their eigenvalues and eigenvectors. It turns out that additional data, which we call boundary data, are needed in order to actually implement the reconstruction. Once this is done we can extract from them basic information about the Neumann coefficients matrices of the ghost three strings vertex. The main results of our paper are the study of the spectral properties of the infinite matrices S in the b c ghost bases, the reconstruction recipe for such infinite matrices n − (which is an interesting result in itself) and the evidence for the existence of the three– strings vertex we need for the ghost sector in the natural normal ordering. The paper is organized as follows. In section 2, which is essentially pedagogical, we present an example of three strings vertex which is not the one we are looking for as it cannot be diagonalized in the weight 2 basis, but has all the other good properties we expect of the true vertex. This example also illustrates the problems one comes across in constructing a ghost three strings vertex. In section 3 we make contact with the results of [1] and give a more detailed proof that the squeezed states in the midterm of (1.1) have the same eigenvalue as the ghost wedge states in the oscillator formalism. We clarify that this is not enough to prove that (1.1) holds, and, in section 4, we show where the problem lies and propose a new heuristic formula for the reconstruction of infinite lame matrices. Finally section 5 is devoted to our conclusions. Three Appendices contain details of the calculations needed in the text. In particular Appendix C presents a new proof of the fundamental eq.(3.5). Notation. Any infinite matrix we meet in this paper is either square short or long legged, or lame. In this regard we will often use a compact notation: a subscript will s – 2 – represent an integer label n running from 2 to , while a subscript will represent a label l ∞ running from 1 to + . So Y ,Y will denote square short and long legged, respectively; ss ll − ∞ Y ,Y will denote short–long and long–short lame matrices, respectively. With the same sl ls meaning we will say that a matrix is (ll),(ss),(sl) or (ls). In a similar way we will denote by V and V a short and long infinite vector, to which the above matrices naturally apply. s l Moreover, while n,m represent generic matrix indices, at times we will use n¯,m¯ to stress that they are short, i.e. n¯,m¯ 2. ≥ 2. The three strings vertex This section is mostly pedagogical. We would like here to explain what are the problems with defining a three strings vertex for the ghost sector that fits the purposes of proving eq.(1.1) The first problem we have to face is normal ordering. We will have in mind two main cases of normal orderings, those we have called natural and conventional normal ordering in [1]. The former is the obvious normal ordering required when the vacuum is 0 , the latter is instead appropriate to the vacuum state c 0 . A consistent vertex for 1 | i | i conventional normal ordering exists, is the one explicitly computed by Gross and Jevicki, [18], who used the vacuum c c 0 (for general problems connected with the ghost sector, 0 1 | i see [26, 33, 32, 37]). But it is not what we need in the natural normal ordering case. A second problem is generated by the ghost insertions, which are free and there is no a priori principle to fix them. We know however that a certain number of conditions should be satisfied. One is BRST invariance of the three strings vertex. This is unfortunately hard to translate into a practical recipe for construction. Other conditions, i.e. cyclicity, bpz– compatibility and commutativity of the Neumann coefficients matrices are more useful from a constructive point of view. In the sequel we will consider a definite example. Even though it turns out not to be the right vertex we are looking for, it will allow us to illustrate many questions which would sound rather abstruse in the abstract. To start with we first recall the relevant anti–commutator and bpz rules [c ,b ] = δ , bpz(c )= ( 1)nc , bpz(b )= ( 1)nb n m + n+m,0 n −n n −n − − − The we define the state ˆ0 = c c c 0 , where 0 is the SL(2,R)–invariant vacuum, the −1 0 1 | i | i | i tensor product of states ωˆ = ˆ0 ˆ0 0 (2.1) 123 1 2 3 h | h | h | h | carrying total ghost number 6, and ω = 0 0 ˆ0 (2.2) 123 1 2 3 | i | i | i | i carrying total ghost number 3. They satisfy ωˆ ω = 1. Finally we write down the 123 123 h | i general form of the three strings vertex 3 ∞ Vˆ = ωˆ eEˆ, Eˆ = c(r)Vˆrsb(s) (2.3) 3 123 n nm m h | K h | − r,s=1n,m X X – 3 – The dual vertex is 3 ∞ V = eE ωˆ E = c(r)†Vrsb(s)† (2.4) 3 123 n nm m | i K | i r,s=1n,m X X Therangeofm,nisnotspecified. However, forreasonsthatwillbecomeclear later, we would like to interpret the matrices Vˆrs and Vrs as square long–legged matrices (ll). But, nm nm as soon as we try to evaluate, for instance, contractions like Vˆ ω in order to compute 3 123 h | i the constant , a problem arises linked to the presence in the exponent (2.3) of couples of K conjugate operators c ,b , c ,b and c ,b . In order to appreciate this problem let us 0 0 −1 1 1 −1 consider the simple case of ec0V00b0. Interpreting this expression literally one gets 1 ec0V00b0 = 1+c V b + c V b c V b +... 0 00 0 0 00 0 0 00 0 2 1 = 1+c (V + V2 +...)b = 1+c (eV00 1)b (2.5) 0 00 2 00 0 0 − 0 It follows that, when inserted in Vˆ ω a term like this does not yield 1, as one would 3 123 h | i expect. Moreover if, instead of the single zero mode we have considered here for simplicity, we had three, the result would be even more complicated. All this is not natural. Let us recall that the meaning of Vˆrs (see [20] and below) is the coefficient of the monomial nm zm+1wn−2 in the expansion of Vˆ R(c(s)(z)b(r)(w))ω in powers of z and w (with op- 3 123 h | | i posite sign). Therefore interpreting the exponentials in (2.3) as in (2.5) is misleading. It is clear that what they really mean is something else. To adapt the oscillator formalism to the desired meaning we proceed as follows. † Let us introduce new conjugate operators η ,ξ , a = 1,0,1, in addition to c ,b , a a n m − such that † [η ,ξ ] = δ (2.6) a b + ab and they anticommute with all the other oscillators. Moreover we require them to satisfy η 0 = 0, 0ξ† = 0 (2.7) a a | i h | while 0η = 0, ξ† 0 = 0 (2.8) a a h | 6 | i 6 † Now let us replace in the exponent of (2.3) c with η (but not c in the exponent of (2.4) a a a † † † withη )andb intheexponentof(2.4)withξ (butnotb intheexponentof(2.3)withξ – a a a a a infactc† andb willnotbeneeded). Withtheserules Vˆ ω = straightforwardly. The a a 3 123 h | i K matrices Vˆrs and Vrs are naturally square long legged. The interpretation of Vˆrs as the nm nm nm negative coefficient of order zm+1 and wn−2 in the expansion of Vˆ R(c(s)(z)b(r)(w))ω 3 123 h | | i in powers of z and w, remains valid provided one replaces b(r)†,b(r)†,b(r)† in b(r)(w) with −1 0 1 (r)† (r)† (r)† ξ ,ξ ,ξ . −1 0 1 † † We stress again that the substitution of c with η and b with ξ is dictated by the a a a a requirement of consistency of the interpretation of the Neumann coefficient as expansion coefficients of the b-c propagator. – 4 – 2.1 Ghost Neumann coefficients and their properties It is time to go to a concrete example. To this end one has to explicitly compute Vˆrs and nm Vrs in (2.3,2.4). Themethod is well–known: one expresses the propagator with zero mode nm insertions c(z)b(w) in two different ways, first as a CFT correlator and then in terms ≪ ≫ of Vˆ and one equates the two expressions after mapping them to the disk via the maps 3 f (z )= α2−if(z ), i = 1,2,3 (2.9) i i i where 1+iz 2 f(z) = 3 (2.10) 1 iz (cid:16) − (cid:17) 2πi Here α = e 3 is one of the three third roots of unity. However this recipe leaves several uncertainties due especially to the ghost insertions. For concreteness in Appendix A we make a specific choice of these insertions, in a way the simplest one: we set the insertions at infinity. Even so there remain some uncertainties which we fix by requiring certain properties, in particular cyclicity, consistency with the bpz operation and commutativity of the twisted matrices of Neumann coefficients (the motivation for the latter will become clear further on). With this (arbitrary) choice, the ghost Neumann coefficients worked out in Appendix A satisfy the following set of properties: cyclicity • Vˆrs =Vˆr+1,s+1, (2.11) nm nm bpz consistency • ( 1)n+mVrs = Vˆrs (2.12) nm nm − commutativity • Its meaning is the following. Defining X = CˆVrr, X+ = CˆV12,X− = CˆV21, we have XrsXr′s′ = Xr′s′Xrs (2.13) for all r,s,r′,s′. In addition we have X +X+ +X− = 1 (2.14) and X+X− = X2 X, X2+(X+)2+(X−)2 = 1 (2.15) − It should be stressed that all the Xrs matrices are (ll). – 5 – 2.2 Formulas for wedge states Our next goal is to define recursion relations for the ghost wedge states. To start with we define the star product of squeezed ghost states of the form S = exp c†Sb† 0 (2.16) | i N | i (cid:16) (cid:17) We notice that since the vacuum is 0 we are implicitly referring to the natural normal | i ordering. The star product of two such states S and S is the bpz of the state 1 2 | i | i Vˆ S S (2.17) 3 1 1 2 2 h || i | i However this formula needs some specifications. We remark that the problem pointed out above, linked to the presence of couples of conjugate oscillators in the exponents, is present both in (2.16) and (2.17). We solve it as we did in section 2, with the help of additional † oscillators η ,ξ . We interpretfor instance(2.16) as follows. We replace thenewoscillators a b in it as in section 2, then we exploit the anticommutativity properties of the latter to move † † them to the right and apply them to 0 , then we substitute back b in the place of ξ . The a a | i † upshot of this operation is that no b oscillator will survive and the state (2.16) takes the a form S = exp c†S b† 0 (2.18) n nm m | i N | i ! n=−1m=2 X X That is, the matrix S in the exponent is lame ls. This is the precise meaning we attach nm to (2.16). Let us notice that the bpz dual expression of (2.18) is S = 0 exp cCˆSCˆb (2.19) h | Nh | − (cid:16) (cid:17) The matrix S here is ll. After this specification let us define the star product of S and S . Let us recall the 1 2 | i | i three strings vertex (2.3,2.4). Remembering the discussion before (2.18) we conclude that Vˆrs is sl for r = 1,2 and ll for r = 3, while Vrs is ls for r = 1,2 and ll for r = 3. nm nm In evaluating this product we will have to evaluate vev’s of the type ˆ0 exp cFb+cµ+λb exp c†Gb†+θb†+c†ζ 0 (2.20) h | | i (cid:16) (cid:17) (cid:16) (cid:17) Here we are using an obvious compact notation: F,G denotes matrices F ,G , and nm nm λ,µ,θ,ζ are anticommuting vectors λ ,µ ,θ ,ζ . We expect the result of this evaluation n n n n to be ˆ0 exp(cFb+cµ+λb)exp c†Gb†+θb†+c†ζ 0 h | | i (cid:0) (cid:1) = det(1+FG) exp θ 1 Fζ λ 1 Gµ θ 1 µ+λ 1 ζ (2.21) − 1+FG − 1+GF − 1+FG 1+GF (cid:16) (cid:17) In order for this formula to hold in (2.20) the operator denoted b,c must be creation operators with respect to ˆ0 and annihilation operators with respect to the 0 vacuum. h | | i – 6 – Viceversatheoscillators denotedc†,b† mustbeallcreationoperatorwithrespectto 0 , and | i annihilation operators with respect to ˆ0 . But this is precisely what happens if we assume h | the definition (2.18) for the squeezed states and (2.3) for the vertex with the summation † over n starting from 2 (which is consistent with the interpretation by means of ξ and η , a a as before (2.18)). Therefore it is correct to use formulas like (2.21) in order to evaluate the star product (2.17), but in this case the matrices F and G will be lame (ls or sl as the case be), while analogous considerations apply to the vectors λ,µ,θ,ζ (λ,ζ are long vectors, while µ,θ are short). The star product of two squeezed states like (2.16) is S ⋆ S = S 1 2 12 | i | i | i where the state in the RHS has the same form as (2.16), with the matrix S replaced by S = CˆT . The latter is given by the familiar formula 12 12 1 X− T = X +(X+,X−) Σ (2.22) 12 1 Σ 12 X+ 12 ! − V where CˆS 0 X X+ 1 Σ = , = (2.23) 12 0 CˆS V X− X 2! ! The normalization of S is given by 12 | i = det(1 Σ ) (2.24) 12 1 2 12 N N N −V Notice that in this formula the four matrices in Σ are ss. 12 V These expressions are well defined. However, since they are expressed in terms of lame matrices we cannot operate with them in the same way we usually do with the analogous matrices of the matter sector. For that one needs the identities proved in the previous section, which are only valid for long square matrices. Luckily in the case of the wedge states it is possible to overcome this difficulty. When computing a star product we would like to be able to apply the formulas of subsection 2.1, which are expressed in terms of long square matrices. To this end we would like (2.21) to be expressed in terms of long square matrices, rather than of lame matrices. This is possible at the price of some modifications. † Let us introduce the new conjugate operators η ,ξ , a = 1,0,1, as above, see a a − † † (2.6,2.7,2.8) and let us replace in (2.21) c (but not c ) with η and b (but not b ) with a a a a a † ξ . Then in the RHS long square matrices and long vectors will feature (instead of lame a matrices and short or long vectors). In the sequel we will use (2.21) in this sense. Such modifications of course are not for free. We have to justify them 1. We will show later on that in the case of the wedge states such a move is justified. 1In the previous cases the introduction of the new oscillators was simply an auxiliary tool to help us interpretsuchformulasas(2.18). Wecouldhavedonewithoutthembyad hocdefinitions. Butnowweare tampering with vev’s,therefore we haveto makesure that we donot modify anythingessential. – 7 – Once this is done the calculation of the star product works smoothly without any sub- stantialdifferencewithrespecttothemattercase. Theformulasarethesameeqs.(2.22,2.23) and (2.24) above, but expressed in terms of long square matrices to which we can apply the identities of subsection 2.1. This allows us to treat the ghost squeezed states in a way completely similar to the matter squeezed states. Of course it remains for us to comply with the promise we made of showing that we are allowed to use long square matrices. The wedge states are now defined to be squeezed states n S that satisfy the n | i ≡ | i recursive star product formula n ⋆ m = n+m 1 (2.25) | i | i | − i This implies that the relevant matrices T = CˆS satisfy the recursion relation n n X (T +T )X +T T n m n m T = − (2.26) n+m−1 1 (T +T )X +T T X n m n m − or 1 T n T = X − , (2.27) n+1 1 T X n − and the normalization constants are given by = det(1 T X) (2.28) n+1 n n N N K − These relations are derived under the hypothesis that T and X,X+,X− commute and by n using the identities of subsection 2.1. The solution to (2.27) is well–known, [36, 37]. We repeat the derivation in order to stress its uniqueness. We require that 2 coincide with | i the vacuum 0 , both for the matter and the ghost sector2. | i This implies in particular that T = 0 and = 1, which entails from (2.27) that 2 2 N T = X, T = X , etc. That is T is a uniquely defined function of X. But X can be 3 4 1+X n uniquely expressed in terms of the sliver matrix T T X = (2.29) T2 T +1 − a formula whose inverse is well–known, [34, 21] 1 T = 1+X (1 X)(1+3X) (2.30) 2X − − (cid:16) p (cid:17) Therefore T can be expressed as a uniquely defined function of T. Now consider the n formula T +( T)n−1 T = − n 1 ( T)n − − 2It is worth recalling that ourpurpose in thispaper is to complete the proof started in [1] of (1.1) e−n−22“L(0g)+L0(g)†”|0i=Nnec†Snb†|0i≡|ni that is, that the LHS does represent the ghost wedge states. In this light the requirement that the wedge state |ni with n=2 coincide with thevacuum state is natural. – 8 – It satisfies (2.27) as well as the condition T = 0, therefore it is the unique solution to 2 (2.27) we were looking for. So far the states n have been defined solely in terms of the three strings vertex. One | i might ask what is their connection with the wedge states defined as surface states, [25, 27, 35, 2]. This connection can be established: it can be shown that, with the appropriate insertion of the zero modes, the surface wedge matrix S is actually Vrr, i.e. T = X. 3 3 It is simple to see that similarly (2.28) has a unique solution satisfying = 1 and 2 N = . 3 K N 2.3 Commutation relations with K 1 What we have done so far is all very good, but the concrete example of vertex constructed in Appendix A is only academical, as the following remark shows. In [1] we diagonalized the LHS of 1.1 on the basis of weight 2 differentials, in which the operator K is diagonal. 1 In order to be able to compare this result with the wedge states defined above we have to make sure that also the matrices T ,X,X ,X can be diagonalized in the same basis. In n + − this subsection we will discuss this problem. Let us recall the definition of K : 1 K = c†G b + b†H c 3c b (2.31) 1 p pq q p¯ p¯q¯ q¯ 2 −1 − p,q≥−1 p,q≥2 X X where G = (p 1)δ +(p+1)δ , pq p+1,q p−1,q − H = (p¯+2)δ +(p¯ 2)δ (2.32) p¯q¯ p¯+1,q¯ p¯−1,q¯ − G is a squarelong–legged matrix and H a squareshort–legged one. In the common overlap we have G = HT. We notice immediately that K annihilates the vacuum 1 K 0 = 0 (2.33) 1 | i What is important for us is that the action of K commutes with the matrices we want to 1 diagonalize. Now let T = CˆS , where S is the matrix of the squeezed state representing n n n n . We have seen that T can be either lame or square (ll). Since we want to diagonalize n | i (2.27) we must consider the second alternative. But in order to arrive at square (ll) matri- ces, at the beginning of this section we introduced into the game the conjugate oscillators † η ,ξ , a = 1,0,1. Therefore, to be consistent, they must appear also in the oscillator a a − representation of K . This can be done as follows. 1 We write down K as 1 K = c†G b + b†H c (2.34) 1 p pq q p pq q p,q≥−1 p,q≥−1 X X where G and H have the same expression as before, but now also H is square long legged and H = GT. What is important is that in the expression b†Hc we understand that b† is a – 9 –